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Digital Image Processing

Digital Image Processing. Chapter 4: Image Enhancement in the Frequency Domain. Background. The French mathematiian Jean Baptiste Joseph Fourier Born in 1768 Published Fourier series in 1822 Fourier ’ s ideas were met with skepticism Fourier series

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Digital Image Processing

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  1. Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain

  2. Background • The French mathematiian Jean Baptiste Joseph Fourier • Born in 1768 • Published Fourier series in 1822 • Fourier’s ideas were met with skepticism • Fourier series • Any periodical function can be expressed as the sum of sines and/or cosines of different frequencies, each multiplied by a different coefficient

  3. Fourier transform • Functions can be expressed as the integral of sines and/or cosines multiplied by a weighting function • Functions expressed in either a Fourier series or transform can be reconstructed completely via an inverse process with no loss of information

  4. Applications • Heat diffusion • Fast Fourier transform (FFT) developed in the late 1950s

  5. Introduction to the Fourier Transform and the Frequency Domain • The one-dimensional Fourier transform and its inverse • Fourier transform • Inverse Fourier transform

  6. Two variables • Fourier transform • Inverse Fourier transform

  7. Discrete Fourier transform (DFT) • Original variable • Transformed variable

  8. DFT • The discrete Fourier transform and its inverse always exist • f(x) is finite in the book

  9. Sines and cosines

  10. Time domain • Time components • Frequency domain • Frequency components

  11. Fourier transform and a glass prism • Prism • Separates light into various color components, each depending on its wavelength (or frequency) content • Fourier transform • Separates a function into various components, also based on frequency content • Mathematical prism

  12. Polar coordinates • Real part • Imaginary part

  13. Magnitude or spectrum • Phase angle or phase spectrum • Power spectrum or spectral density

  14. Samples

  15. Some references • http://local.wasp.uwa.edu.au/~pbourke/other/dft/ • http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm

  16. Examples • test_fft.c • fft.h • fft.c • Fig4.03(a).bmp • test_fig2.bmp

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